Combining what everybody has said, I can tell you that the following is the guidelines that should be followed:
1) Do not ever offer a draw on your opponent's time. If it bothers your opponent enough, that actually can be reported to the director.
2) Do not ever offer a draw before you make your move. You might suddenly recognize a win, but guess what? You're stuck offering it, and your opponent has until he physically makes a move to accept or decline it, and can REQUIRE you to move before he decides!
Instead, proper procedure is to make your move, announce that you are offering a draw, and then hit your clock.
3) Do not offer repetitive draws. If your opponent rejects the draw, then one of two things needs to happen. The first is that he has offered one back to you. So if you offered a draw after your 36th move, he declines, he offers one back at move 39, you decline, there is nothing wrong with you offering it back at move 41, even though you offered one just 5 moves ago. The second is that the position has changed substantially, or a complete lack of progress has been proven. There is ABSOLUTELY NOTHING that says you can only offer a draw twice. Let's say a draw wins the tournament for you. You offer a draw at move 2. Opponent declines. You get to a pawn down Rook and Pawn ending, you offer again at move 41, he declines. You then 12 moves later get to Philidor's Draw, and have clearly demonstrated it for the last 3 or 4 moves, just toggling your Rook on the 6th (3rd if you are White) rank. There is nothing wrong with offering a third draw there.
4) DO NOT EVER hold out your hand when you offer a draw. If your opponent knows all the technicalities, he can shake your hand, say good game, and put 1 for himself, and 0 for you. Holding out your hand to your opponent after the game has been started (i.e. at least 1 move by each player) is interpreted as a RESIGNATION! Before the game begins, or late arrival where at least one player hasn't moved yet, like if Black shows up late, White starts his own clock, moves d4, and pushes his clock, Black shows up 12 minutes later, he can put his hand out to wish good luck. After the first move by Black, holding your hand out can be taken as a resignation and scored immediate!
5) As for online games, ChessClub.com is better than here. The primary interface it uses (I think it's called "Dasher", but "Blitzin" also did the same thing) announces "Draw Offered", so you don't have to look and pause. I only accept online draws in clearly drawn positions and make draw claims on 3-fold. Online chess ratings are garbage anyway. Disconnects due to power outages, blitzing people out of time where you win KRPPP vs KRNNPPPPP, etc. My blitz rating here is in the low 1600s. My over the board rating, the one that really counts and identifies your true skill, is 2048 for USCF and 2057 for FIDE.
Also, some of the dumbest old wive's tales I hear, which is complete bogus, are the following:
1) Don't offer a draw to a higher rated player - UHM, BULLSH*T, I'll offer a draw to whoever I want to offer a draw to.
2) Treat higher rated players like they are your elders - UHM, again, BULLSH*T, Higher rated players, as far as I'm concerned, are not to be treated with any more special treatment than lower rated players. If you get White, you go first. If you think the position is drawn, and want to offer a draw, offer it. You want to play the god-awful London System, go ahead and play it. Just because you face a 2600 player doesn't mean you have to act more manly. If you have Black, and want the clock on your left, put it on your left. I could care less if he's 2600 and requesting the clock on the other side. If he has Black, I am more than willing to honor his request. He has that right. If I get Black, FORGET IT!
If you make a move, it counts as a draw refusal. You don't have to go click the "decline" button.
I know that but when the "draw" button lights up, it makes me pause for a second and in a blitz game with the time running out, that second is critical.